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<h1 class="ltx_title ltx_title_section">PSF model: Symmetric 2D Gaussian function</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">A common approximation of the real PSF is a symmetric two-dimensional
Gaussian function given by the formula</p>
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<div id="Sx1.p2" class="ltx_para">
<table id="Sx1.E1" class="ltx_equation">

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<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-24px" src="mi/mi153.png" width="466" height="57" alt="\mathrm{PSF_{G}}\left(x,y\mid\boldsymbol{\theta}\right)=\frac{\theta_{N}}{2\pi%
\theta_{\sigma}^{2}}\exp{\left(-\frac{\left(x-\theta_{x}\right)^{2}+\left(y-%
\theta_{y}\right)^{2}}{2\theta_{\sigma}^{2}}\right)}+\theta_{b}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
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<p class="ltx_p">where <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi154.png" width="118" height="21" alt="\mathrm{PSF_{G}}\left(x,y\mid\boldsymbol{\theta}\right)"> gives
the expected photon count at the integer pixel position <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi60.png" width="45" height="21" alt="\left(x,y\right)">
for a vector of parameters <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-7px" src="mi/mi113.png" width="167" height="22" alt="\boldsymbol{\theta}=\left[\theta_{x},\theta_{y},\theta_{\sigma},\theta_{N},%
\theta_{b}\right]">.
The entries of the vector <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi56.png" width="15" height="16" alt="\boldsymbol{\theta}"> are as follows: <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi67.png" width="21" height="19" alt="\theta_{x}">
and <img id="Sx1.p2.m6" class="ltx_Math" style="vertical-align:-7px" src="mi/mi68.png" width="21" height="21" alt="\theta_{y}"> are the sub-pixel molecular coordinates, <img id="Sx1.p2.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi92.png" width="22" height="19" alt="\theta_{\sigma}">
is the imaged size of the molecule, <img id="Sx1.p2.m8" class="ltx_Math" style="vertical-align:-5px" src="mi/mi63.png" width="26" height="19" alt="\theta_{N}"> corresponds to the
total number of photons emitted by the molecule, and <img id="Sx1.p2.m9" class="ltx_Math" style="vertical-align:-5px" src="mi/mi66.png" width="20" height="19" alt="\theta_{b}">
corresponds to the background signal level.</p>
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<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

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<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
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</li>
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<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
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<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
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</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="LocalizationUncertainty.html" title="" class="ltx_ref">Localization uncertainty</a></p>
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</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="CrowdedField.html" title="" class="ltx_ref">Multiple-emitter fitting analysis</a></p>
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